#503153

order	512
nilpotency class	4
isoclinism	isoclinic class of #501262
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1133 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(2,3)(5,6)(7,13)(8,10)(9,16)(11,30)(12,18)(14,17)(15,25)(19,23)(20,33)(21,36)(22,35)(24,29)(26,28)(27,32)(31,34);(1,2,4,3)(5,7,12,23)(6,13,18,19)(8,24,15,33)(9,27,28,21)(10,29,25,20)(11,31,14,22)(16,32,26,36)(17,35,30,34);(5,8,25,18,11,26,28,30,12,15,10,6,14,16,9,17)(7,19,27,24,22,35,29,32,23,13,21,33,31,34,20,36);(1,3,4,2)

Polycyclic presentation:

2^9|b^2=ei,c^2=fh,d^2=e,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=ghi,h^b=hi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410349);